Three-Dimensional CR Submanifolds in $S^6(1)$ with Umbilical Direction Normal to $\mathcal{D}_3$

Year 2021, Volume: 14 Issue: 1, 125 - 131, 15.04.2021

Miroslava Antic Djordje Kocic

https://doi.org/10.36890/iejg.790910

Abstract

It is well known that the sphere $S^6(1)$ admits an almost complex structure $J$ which is nearly K\"{a}hler. A submanifold $M$ of an almost Hermitian manifold is called a CR submanifold if it admits a differentiable almost complex distribution $\mathcal{D}_1$ such that its orthogonal complement is a totally real distribution. In this case the normal bundle of the submanifold also splits into two distributions $\mathcal{D}_3$, which is almost complex, and a totally real complement. In the case of the proper three-dimensional CR submanifold of a six-dimensional manifold the distribution $\mathcal{D}_3$ is non-trivial. Here, we investigate three-dimensional CR submanifolds of the sphere $S^6(1)$ admitting an umbilic direction orthogonal to $\mathcal{D}_3$ and show that such submanifolds do not exist.  

Keywords

CR submanifolds, umbilical direction, nearly Kähler 6-sphere

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